Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $q = \dfrac{t^2 - 3t - 40}{9t - 72} \div \dfrac{t^2 + 5t}{t - 8} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{t^2 - 3t - 40}{9t - 72} \times \dfrac{t - 8}{t^2 + 5t} $ First factor the quadratic. $q = \dfrac{(t + 5)(t - 8)}{9t - 72} \times \dfrac{t - 8}{t^2 + 5t} $ Then factor out any other terms. $q = \dfrac{(t + 5)(t - 8)}{9(t - 8)} \times \dfrac{t - 8}{t(t + 5)} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac{ (t + 5)(t - 8) \times (t - 8) } { 9(t - 8) \times t(t + 5) } $ $q = \dfrac{ (t + 5)(t - 8)(t - 8)}{ 9t(t - 8)(t + 5)} $ Notice that $(t - 8)$ and $(t + 5)$ appear in both the numerator and denominator so we can cancel them. $q = \dfrac{ \cancel{(t + 5)}(t - 8)(t - 8)}{ 9t(t - 8)\cancel{(t + 5)}} $ We are dividing by $t + 5$ , so $t + 5 \neq 0$ Therefore, $t \neq -5$ $q = \dfrac{ \cancel{(t + 5)}\cancel{(t - 8)}(t - 8)}{ 9t\cancel{(t - 8)}\cancel{(t + 5)}} $ We are dividing by $t - 8$ , so $t - 8 \neq 0$ Therefore, $t \neq 8$ $q = \dfrac{t - 8}{9t} ; \space t \neq -5 ; \space t \neq 8 $